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Effects of weak nonlinearity on the dispersion relation and frequency band-gaps of aperiodic Bernoulli–Euler beam

Sorokin, Vladislav S.; Thomsen, Jon Juel

Published in:Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Link to article, DOI:10.1098/rspa.2015.0751

Publication date:2016

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Citation (APA):Sorokin, V. S., & Thomsen, J. J. (2016). Effects of weak nonlinearity on the dispersion relation and frequencyband-gaps of a periodic Bernoulli–Euler beam. Proceedings of the Royal Society A: Mathematical, Physical andEngineering Sciences, 472(2186), [20150751]. https://doi.org/10.1098/rspa.2015.0751

https://doi.org/10.1098/rspa.2015.0751

https://orbit.dtu.dk/en/publications/86776e74-2283-4a94-a859-35676391a967

https://doi.org/10.1098/rspa.2015.0751

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ResearchCite this article: Sorokin VS, Thomsen JJ.2016 Effects of weak nonlinearity on thedispersion relation and frequency band-gapsof a periodic Bernoulli–Euler beam. Proc. R.Soc. A 472: 20150751.http://dx.doi.org/10.1098/rspa.2015.0751

Received: 29 October 2015Accepted: 13 January 2016

Subject Areas:mechanical engineering, appliedmathematics, wave motion

Keywords:elastic wave propagation, dispersion relation,frequency band-gaps, weak nonlinearity,periodic Bernoulli–Euler beam, method ofvarying amplitudes

Author for correspondence:Vladislav S. Sorokine-mail: [email protected]

Effects of weak nonlinearity onthe dispersion relation andfrequency band-gaps of aperiodic Bernoulli–Euler beamVladislav S. Sorokin1,2 and Jon Juel Thomsen1

1Department of Mechanical Engineering, Technical University ofDenmark, Nils Koppels Allé, Building 404, 2800 Kongens Lyngby,Denmark2Institute of Problems in Mechanical Engineering RAS, V.O., Bolshojpr. 61, St Petersburg 199178, Russia

VSS, 0000-0003-4457-7066

The paper deals with analytically predicting theeffects of weak nonlinearity on the dispersion relationand frequency band-gaps of a periodic Bernoulli–Euler beam performing bending oscillations. Twocases are considered: (i) large transverse deflections,where nonlinear (true) curvature, nonlinear materialand nonlinear inertia owing to longitudinal motionsof the beam are taken into account, and (ii) mid-plane stretching nonlinearity. A novel approach isemployed, the method of varying amplitudes. As aresult, the isolated as well as combined effects ofthe considered sources of nonlinearities are revealed.It is shown that nonlinear inertia has the mostsubstantial impact on the dispersion relation of a non-uniform beam by removing all frequency band-gaps.Explanations of the revealed effects are suggested, andvalidated by experiments and numerical simulation.

1. IntroductionThe analysis of the behaviour of linear periodicstructures can be traced back over 300 years to SirIsaac Newton [1], but until Rayleigh’s work [2] thesystems considered were lumped masses joined bymassless springs. Much attention was given to thistopic in the twentieth century (e.g. [1,3,4]). In recentyears, the topic has experienced rising interest [5–9]. Anessential feature of periodic structures is the presenceof frequency band-gaps, i.e. frequency ranges in whichwaves cannot propagate. The determination of band-gaps and the corresponding attenuation levels is an

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important practical problem [3–9]. A large variety of analytical methods have been developedto solve this problem, most of them based on Floquet theory [1]; this holds, for example, for theclassical Hill’s method of infinite determinants [10,11] and the method of space harmonics [12].However, application of these for nonlinear problems is impossible or cumbersome, becauseFloquet theory is applicable for linear systems only. Thus, the nonlinear effects arising in periodicstructures have not yet been fully discovered, while, at the same time, applications may requirethe effects of nonlinearity on the structural response to be accounted for. Only a few papershave been devoted to this topic, most of them either considering lumped-parameter models, i.e.discrete periodic mass-spring chains [13–15], or implying a certain discretization of continuousperiodic structures, e.g. by the Galerkin weighted residuals approach [16] or finite elements [17].Khajehtourian & Hussein [18] consider longitudinal wave motion in a one-dimensional elasticmetamaterial, which is a thin uniform rod with periodically attached local resonators, whereasAbedinnasab & Hussein [19] studied the dispersion relations for a uniform rod and Bernoulli–Euler beam under finite deformation, with the effects of nonlinearities accounted for. The latterproblem represents a special case of the one covered in this paper, in the limit of zero modulation.

This article deals with analytically predicting the dynamic responses for a nonlinearcontinuous elastic periodic structure without employing system discretization. Specifically,the effects of weak nonlinearity on the dispersion relation and frequency band-gaps of aperiodic Bernoulli–Euler beam performing bending oscillations are analysed. Various sources ofnonlinearity are considered: nonlinear (true) curvature, nonlinear inertia owing to longitudinalbeam motions, nonlinear material and the nonlinearity associated with mid-plane stretching.A novel analytical approach is employed, the method of varying amplitudes (MVA) [20,21]. Thisapproach is inspired by the method of direct separation of motion (MDSM) [22,23], and maybe considered a natural continuation of the classical methods of harmonic balance [11] andaveraging [24–26]. It implies a solution in the form of a harmonic series with varying amplitudes;however, in contrast to the asymptotic methods, the amplitudes are not required to varyslowly. Thus, the approach does not imply separation of variables into slow and fast, whichis the key assumption of the MDSM. It is also strongly related to Hill’s method of infinitedeterminants [1,10,11], and to the method of space harmonics [12].

Possible sources of nonlinearities for a Bernoulli–Euler beam performing bending oscillationshave been discussed in many works; see, for example, [27] and the classical monograph [11].In [11], the main sources were identified as nonlinear stiffness and nonlinear inertia. It wasnoted that the character of the nonlinearity strongly depends on the specific boundary conditionsapplied to the beam. For example, when there is no restriction on the longitudinal motion ofthe beam ends, large deflections are possible, so that nonlinear (true) curvature and nonlinearinertia owing to longitudinal motion of the beam should be taken into account. The effectsof nonlinear material may also be of significance in this case. If both ends of the beam arerestricted from moving in the longitudinal direction, then another source of nonlinearity becomesimportant, namely mid-plane stretching. This nonlinearity appears to be much stronger than theothers [28,29] and influences the beam response at relatively small deflections.

Because, in real structures, boundary conditions affect the character of the nonlinearity,finite structures are to be considered. On the other hand, the analysis of dispersion relationsand frequency band-gaps, in its conventional formulation [1,12], implies considering infinitestructures. The transition from infinite to finite structures and the discussion of the importanceof dispersion relations and band-gaps for finite structures are given in many studies [1,5,7,12,30].Dispersion relations are of utmost importance for finite structures, e.g. because they facilitatedetermining natural frequencies and mode shapes. Frequency band-gaps, in turn, are key featuresof sufficiently long structures for which waves from band-gap ranges attenuate strongly beforereaching the boundaries.

Section 2 is concerned with the formulation of the governing equations of transverse motionsof the beam and their brief analysis. In §3, the equations are solved by the MVA; §4 presentsthe obtained dispersion relations and reveals the effects of nonlinearities on the frequency band-gaps. Section 5 is concerned with the discussion as well as with the experimental and numericalvalidation of the results.



3


...................................................

2. Governing equations

(a) Case A: beam unrestricted longitudinally; relatively large deflections possibleConsider the case with no restriction on the longitudinal motions of the beam, when relativelylarge transverse deflections are possible. This case corresponds to, for example, the clamp-freebeam schematically shown in figure 1a, with the kinematics of the beam element also presented.The internal bending moment of a Bernoulli–Euler beam with spatially varying properties isdefined by

M(x, t) = EI(x)κ , (2.1)

where I is the moment of inertia of the cross section, E is Young’s modulus of the beam material, xis the axial coordinate along the deformed beam and κ(x, t) is the nonlinear (true) curvature [11,27]

κ(x, t) = w′′√1 − (w′)2

≈(

1 + 12

(w′)2)

w′′. (2.2)

Here, w(x, t) is the transverse displacement of the beam at time t and axial coordinate x, primesdenote derivatives with respect to x and the approximation in (2.2) assumes finite, but notvery large, rotations w′ [11,28]. Taking into account the effects of nonlinear material [11], werewrite (2.1) as

M(x, t) = EI(x)[κ − βnκ3], (2.3)

where the coefficient βn defines the nonlinearity of the beam material stress–strain relation; formost materials, this nonlinearity is symmetric and of a ‘softening’ type, i.e. there is no quadraticterm in (2.3), and βn > 0 [11]. Inserting (2.2) into (2.3) and keeping the nonlinearities to third order,we obtain

M(x, t) = EI(x)[

1 + 12

(w′)2 − βn(w′′)2]

w′′. (2.4)

The longitudinal displacement of the beam cross section owing to transverse deflections is [11,27]

u(x, t) = −12

∫(w′)2 dx. (2.5)

With no restriction on the longitudinal boundary motions, Newton’s second law applied in thelongitudinal direction gives

N′ = ρA(x)∂2u

∂ t2, (2.6)

where N is the additional longitudinal force owing to the effects of inertia, and ρA(x) is the beammass per unit length. By integration and with (2.5), one obtains

N(x, t) =∫

ρA(x)∂2u

∂ t2dx = −1

2

∫ρA(x)

∂2

∂ t2

[∫(w′)2 dx

]dx. (2.7)

The beam mass per unit length ρA(x) and (linear) bending stiffness EI(x) are assumed to beperiodically varying in the axial coordinate x

ρA(x) = ρA(x + Θ), EI(x) = EI(x + Θ), (2.8)

where Θ is the period of modulation, and

I(x) = (r(x))2A(x), (2.9)



4


...................................................

x~w~

w~w~ + dw~

w~ + dw~

x~dx~

dx~

dx~

(1+ L) d

x~(b)

(a)

u

u + du

u + du

u

Figure 1. Schematic of the periodic beams under consideration with an illustration of the kinematics of the beam element; (a)case A and (b) case B. (Online version in colour.)

where r(x) is the radius of gyration of the cross-sectional area A(x). Expanding ρA(x) and EI(x) ina Fourier series gives

ρA(x) = ρA0

[1 +

∞∑m=1

χA,m sin(

2π

Θmx + φA,m

)]

and EI(x) = EI0

[1 +

∞∑m=1

χI,m sin(

2π

Θmx + φI,m

)].

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

(2.10)

Our aim is to reveal the principal effects of the nonlinearities on the dispersion relation andfrequency band-gaps of a periodic beam. Consequently, as a first approximation, only thefundamental harmonic is accounted for in (2.10), so that

ρA(x) = ρA0(1 + χA sin(kx + φ))

and EI(x) = EI0(1 + χI sin(kx + φ)),

}(2.11)

where 0 ≤ χA < 1, 0 ≤ χI < 1, k = 2π/Θ , and in the simplest case of constant ρ, E and r, we haveχA = χI. Here, a more general case is considered, where the modulation amplitudes χA and χI arenot required to be equal, though modulations of the beam mass per unit length and stiffness havethe same phase shift φ. According to [1], the approximation similar to (2.11) is valid for predictingat least the lowest two band-gaps of a periodic structure.

The governing equation of transverse motions of the beam is [11,28]

M′′ − (Nw′)′ + ρA(x)∂2w

∂ t2= 0, (2.12)

which assumes waves much longer than the height of the beam, so that the classical Bernoulli–Euler theory holds, and rotary inertia and shear deflections can be ignored. Dissipation is nottaken into account, which is typical [1,9,12] for studying dispersion relations and frequency band-gaps of periodic structures. Inserting (2.4) and (2.11) into (2.12) gives

ρA0(1 + χA sin(kx + φ))∂2w

∂ t2− (Nw′)′

+ EI0

[(1 + χI sin(kx + φ))

(1 + 1

2

(w′)2 − βn

(w′′)2) w′′

]′′= 0. (2.13)

Introducing non-dimensional variables x = kx + φ, t = ωt, and w = kw, where ω = k2√

EI0/(ρA0)is the frequency of waves with length 2π/k propagating in the corresponding uniform beam, (2.13)



5


...................................................

can be rewritten in dimensionless form as

(1 + χA sin x)∂2w∂t2 +

[(1 + χI sin x)

(1 + 1

2

(w′)2 − βn

(w′′)2)w′′

]′′− (

Nw′)′ = 0, (2.14)

where βn = βnk2,

N(x, t) = −12

∫(1 + χA sin x)

∂2

∂t2

[∫ (w′)2 dx

]dx, (2.15)

and primes now denote derivatives with respect to the non-dimensional spatial coordinate x.Because w = kw the effect of the nonlinearities depends not only on the magnitude of the

physical deflections w, but also on the value of k, which is the spatial frequency of modulations, sothat for large k, i.e. for rapidly varying cross section, the effect can be significant even for relativelysmall physical deflections of the beam.

Solutions of (2.14) are sought in the form of a series, with over-bars denoting complexconjugation

w(x, t) = ϕ(x)eiωt + ϕ(x)e−iωt + ϕn(x)e3iωt + ϕn(x)e−3iωt + · · · , (2.16)

which is typical for problems involving oscillations of weakly nonlinear (homogeneous)structures with only symmetric forces being present [31,32]. Only waves with period of thesame order as, or much larger than, the period Θ of modulation are considered, so high-frequency oscillations are outside the scope of this article. In addition, dimensionless deflectionsw are assumed to be finite, but not very large, so that nonlinearities can be considered weak,permitting only the fundamental harmonic in (2.16) to be included. This assumption, in particular,implies that for rapidly varying cross section, i.e. large k, only very small physical deflectionsw are allowed. The simplification, which is validated in §5, also agrees with the low-frequencyapplicability range of the Bernoulli–Euler theory.

Substituting (2.16) and (2.15) into (2.14) and balancing the terms at the fundamental harmonicω, one obtains the governing ordinary differential equation for ϕ(x),

[(1 + χI sin x)

(ϕ′′ + 1

2

(2ϕ′′ϕ′ϕ′ + ϕ′′(ϕ′)2

)− 3βn

(ϕ′′)2 ϕ′′

)]′′− ω2

[(1 + χA sin x)ϕ

+ 2[ϕ′

∫ ((1 + χA sin x)

∫(ϕ′)2 dx

)dx]′]

= 0. (2.17)

Because only waves with a period of the same order as, or much larger than, the period Θ ofmodulation are considered, and owing to the choice of the non-dimensional variables, we haveω = O(1) (which comprises also the case ω � 1).

The integral term in (2.17) represents nonlinear inertia, whereas the term with βn is relatedto nonlinear material, and the remaining nonlinear terms are due to the true measure ofcurvature (2.2). Note that, although βn � 1, the coefficient βn = βnk2 can be of order unity forlarge k.

(b) Case B: mid-plane stretchingNow consider the case when both ends of the beam are restricted from moving in thelongitudinal direction, and mid-plane stretching occurs. This case corresponds to, for example,the clamped–clamped beam schematically shown in figure 1b. To transversely deform such abeam considerably more energy needs to be supplied, because bending is coupled with axialstretching of the beam. Consequently, one can expect transverse deformations to be much smallerthan in case A (§2a), so that the linear measure of curvature and material stress–strain relation canbe adopted, and nonlinear inertia can be neglected [28]. In the absence of external axial forces, the



6


...................................................

assumption regarding longitudinal inertia implies that

N′ = 0, (2.18)

where prime denotes the derivative with respect to x; this means that for any t the longitudinalforce is constant throughout the beam. By Hooke’s law [28,29]

N(x, t) = EA(x)Λ, (2.19)

where Λ(x, t) is the full axial strain

Λ(x, t) =√

(1 + u′)2 + (w′)2 − 1 ≈ u′ + 1

2

(w′)2 , (2.20)

where all variables have the same meaning as for case A and it is assumed that u′ = O((w′)2) � 1,i.e. the axial strain is small. Consequently, by (2.19) and (2.20)

u′ = NEA(x)

− 12

(w′)2 , (2.21)

so that the longitudinal displacement is

u(x, t) =∫ (

NEA(x)

− 12

(w′)2)dx = N

∫1

EA(x)dx − 1

2

∫ (w′)2 dx. (2.22)

Now, by contrast to case A, the beam ends are restricted to move longitudinally, so that u(l) −u(0) = ηl, where η describes a small initial stretch of the beam, and l is the beam length. Imposingthis condition with (2.22) and solving for N gives

N(x, t) = 1

∫l0 1/(EA(x)) dx

(ηl + 1

2

∫ l

0

(w′)2 dx

). (2.23)

Inserting (2.23) into (2.12), adopting the linearized curvature and neglecting the possible effectsof nonlinear material, one obtains

(EI(x)w′′)′′ + ρA(x)

∂2w

∂ t2− 1

∫l0 1/(EA(x))dx

(ηl + 1

2

∫ l

0

(w′)2 dx

)w′′ = 0. (2.24)

Assuming the variation (2.11) of spatial properties, (2.24) becomes

(1 + χA sin x)∂2w∂t2 + [

(1 + χI sin x) w′′]′′− μ

1

∫l+φφ 1/(1 + χA sin x) dx

(ηl + 1

2

∫ l+φ

φ

(w′)2 dx

)w′′ = 0, (2.25)

where all parameters and variables are dimensionless and have the same meaning as in §2a, l = klis the non-dimensional beam length, μ = A0/(I0k2) = (kr0)−2, where r0 is the radius of gyration ofthe corresponding uniform beam, and here again primes denote derivatives with respect to x.

Employing the Bernoulli–Euler theory and considering waves of length much larger than theheight of the beam implies that

λ � r(x) =√

I(x)A(x)

, (2.26)

where λ is the wavelength. Taking into account also that only waves with a period of the sameorder as, or much larger than, the period of property modulation are considered, i.e. λ−1 = O(k),



7


...................................................

one obtains that relation

k−1 �√

I0

A0= r0 (2.27)

should hold true for (2.26) to be satisfied. Consequently, μ � 1, and the nonlinear term in (2.25) ismuch larger than the nonlinear terms in (2.14), illustrating why mid-plane stretching nonlinearityis much stronger than all other nonlinearities considered in §2a.

Searching for a solution to (2.25) in the form (2.16) with only the first harmonic taken intoaccount, which is valid for weak nonlinearity, we obtain equation (2.28) for the new variable ϕ(x)

[(1 + χI sin x) ϕ′′]′′ − μ

1

∫l+φφ 1/(1 + χA sin x)dx

(ηlϕ′′ + ϕ′′

∫ l+φ

φ

ϕ′ϕ′dx + 12ϕ′′

∫ l+φ

φ

(ϕ′)2 dx

)

− ω2(1 + χA sin x)ϕ = 0. (2.28)

By contrast to case A, the beam length l and phase φ are here present in the governing equation forϕ(x), so that the effect of nonlinearity may depend on these parameters. However, it is expectedthat for relatively large l, allowing attenuation of waves from band-gap ranges before reachingthe boundaries, this dependency should vanish.

3. Solution by the method of varying amplitudes

(a) Case A: beam unrestricted longitudinally; relatively large deflections possibleConventional methods for analysing spatially periodic structures, e.g. the classical Hill methodof infinite determinants [10,11] and the method of space harmonics [12], are not applicablefor the problem considered here because they are based on Floquet theory, which is validfor linear systems only. In addition, the governing equations (2.17) and (2.28) are nonlinearintegrodifferential equations involving strong parametric excitation, and solving such equationsby standard asymptotic methods, e.g. the multiple scales perturbation method, is impossible orvery cumbersome [26]. Consequently, a novel approach, the MVA [20,21], is employed. Followingthis approach, a solution to (2.17) is sought in the form of a series of spatial harmonics withvarying amplitudes

ϕ(x) =∞∑

j=−∞bj(x) exp(ijx) = b0(x) + b1(x) exp(ix) + b−1(x) exp(−ix) + · · · , (3.1)

where the complex-valued amplitudes bj(x) are not required to vary slowly in comparison withexp(ijx), so that no restrictions on the solution are imposed. We simply reformulate the problemwith respect to the new variables bj(x). The solution ansatz implied in the MVA, i.e. the choice ofharmonics in (3.1), depends on the parameters of modulation in the equation considered. For thepresent problem, the modulation is sin x = 1

2i (exp(ix) − exp(−ix)), so that ansatz (3.1) is employed.The shift from the original dependent variable ϕ(x) to the new variables bj(x) implies that

additional constraints on these variables should be imposed. With the MVA, the constraints areintroduced in the following way: substitute (3.1) into the governing equation (2.17) and requirethe coefficients of the spatial harmonics involved to vanish identically. As a result, one obtains thefollowing infinite set of differential equations for the amplitudes bj(x)

b′′′′j + 4ijb′′′

j − 6j2b′′j − 4ij3b′

j + (j4 − ω2)bj + i2ω2χA(bj−1 − bj+1)

− i2 χI

(b′′′′

j−1 + 2i(2j − 1)b′′′j−1 −

((2j − 1)2 + 2j(j − 1)

)b′′

j−1

− 2ij(j − 1)(2j − 1)b′j−1 + j2(j − 1)2bj−1



8


...................................................

− b′′′′j+1 − 2i(2j + 1)b′′′

j+1 +(

(2j + 1)2 + 2j(j + 1))

b′′j+1

+ 2ij(j + 1)(2j + 1)b′j+1 − j2(j + 1)2bj+1

)= Nj(b), (3.2)

where j ∈ (−∞, ∞), b = {b0 b1 b−1 b2 b−2 . . .

}T , and Nj(b) are nonlinear functions which are ratherlengthy, and thus not given here. When composing equations (3.2), the relation∫

g(x) exp(ijx) dx = G(x) exp(ijx) (3.3)

has been employed, where G and g are related by

dG(x)dx

+ ijG(x) − g(x) = 0. (3.4)

The approximation of the method is concerned with truncation of the solution series (3.1) andneglecting the corresponding higher-order harmonic terms, similar to the method of harmonicbalance [11,33]. This approximation is valid if the neglected terms in (3.2) are small in comparisonwith those kept, leading to the requirement that the number of terms in the solution seriesemployed is high enough. For the present problem, truncation of the mth harmonic in (3.1) isadequate if

12

∣∣∣m2(m − 1)2χI − ω2χA

∣∣∣� ∣∣∣ω2 − (m − 1)4∣∣∣ , (3.5)

and the involved nonlinearities are weak. Because ω = O(1) and 0 ≤ χA < 1, an additionalrestriction χI � 1 should be imposed to satisfy (3.5) (in fact, it is sufficient to require χI ≤ 0.5;see §5). So only comparatively small modulations of the beam stiffness can be considered by themeans of the method; modulations of the beam mass per unit length, however, can be large.

Equations (3.2) are nonlinear differential equations in b(x). They allow a multitude of solutions,in particular those that can be written in the form

b(x) = bc exp(−iκx). (3.6)

In the linear case, Nj(b) = 0, equations (3.2) have solutions only of the form (3.6), with −iκ beinga root of the characteristic equation of the system (3.2), and bc the associated vector. Taking intoaccount (2.16), (3.1) and (3.6), the solution of the linear counterpart of the initial dimensionlessequation (2.14) may be written as

w(x, t) = F(x) exp (i(ωt − κx)) + cc

and F(x) =∞∑

j=−∞bjc exp(ijx) = b0c + b1c exp(ix) + b−1c exp(−ix) + · · · ,

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(3.7)

where cc denotes complex conjugate terms. This solution obeys Floquet theory [1], because F(x)has the same period as the cross-section modulation, and is similar to the one implied in themethod of space harmonics [12]. It describes a ‘compound wave’ [1] or a ‘wave package’ [12]propagating (or attenuating) in the beam with dimensionless frequency ω and wavenumber κ ,with the relation between ω and κ defining the dispersion relation of the considered periodicstructure, and with real values of κ corresponding to propagating waves and complex values toattenuating waves [1,12].

From (3.7), for the propagation constant p [1,12], describing how a travelling wave changeswhen passing through a single periodic cell, one obtains

p = exp (2πκB) = exp (−i2πκ) , (3.8)

where κB is the Bloch parameter [1,12], which by (3.8) becomes

κB = −iκ , (3.9)

so that real values of κ correspond to propagating waves, and complex values to attenuatingwaves.



9


...................................................

Our aim is to examine the effect of nonlinearities on the beam dispersion relation andfrequency band-gaps. This implies that we are interested in solutions to (3.2) only of theform (3.6), so that the corresponding solution of the initial dimensionless equation (2.14) takesthe form (3.7) describing the propagating (or attenuating) wave with dimensionless frequency ω

and wavenumber κ . Introducing

bjc = �

bjeiθj , j ∈ (−∞, ∞), (3.10)

where�

bj and θj are real-valued constants, and substituting into (3.2) and their complex conjugates,gives

θ2j = θ , θ2j−1 = θ − π

2, j ∈ (−∞, ∞), (3.11)

where θ can take arbitrary values without affecting the resulting algebraic equations for�

bj:

�

bj

((κ − j)4 − ω2

)+ (−1)j ω

2χA

2(

�

bj−1 − �

bj+1) + (−1)j (κ − j)2χI

2

×(

�

bj+1(κ − j − 1)2 − �

bj−1(κ − j + 1)2)

= �

Nj, (3.12)

where j ∈ (−∞, ∞),�

Nj are nonlinear in�

bj and depend on κ , ω, χI, χA and βn. The effect ofnonlinearities on the dispersion relation and frequency band-gaps depends on the magnitudeof transverse deflections w as given by expression (3.7). Taking into account (3.10) and (3.11), weobtain that the spatially averaged amplitude of the beam transverse deflections w is given by

B = 2

√√√√ ∞∑j=−∞

�

b2j . (3.13)

Consequently, the amplitudes�

bj can be compared in magnitude with this value. The algebraicequations (3.12) are then solved numerically for

�

bj and κ as functions of the amplitude B andparameters ω, χI, χA and βn. Thus, the dispersion relation κ = κ(ω) of the considered nonlinearperiodic beam is obtained for various values of the parameters, as will be illustrated in §4, anddiscussed and validated in §5.

(b) Case B: mid-plane stretchingEmploying again the MVA and searching for a solution of (2.28) in the form (2.29), one obtainsthe following infinite set of differential equations for the new variables bj(x):

b′′′′j + 4ijb′′′

j − 6j2b′′j − 4ij3b′

j + (j4 − ω2)bj + i2 ω2χA(bj−1 − bj+1)

− i2 χI

(b′′′′

j−1 + 2i(2j − 1)b′′′j−1 −

((2j − 1)2 + 2j(j − 1)

)b′′

j−1

− 2ij(j − 1)(2j − 1)b′j−1 + j2(j − 1)2bj−1

− b′′′′j+1 − 2i(2j + 1)b′′′

j+1 +(

(2j + 1)2 + 2j(j + 1))

b′′j+1

+ 2ij(j + 1)(2j + 1)b′j+1 − j2(j + 1)2bj+1

)= (S1 + S2)(b′′

j + 2ijb′j − j2bj) + S3(b′′

−j + 2ijb′−j − j2b−j), (3.14)

where

S1 = μ

Hηl, S2 = μ

H

∫ l+φ

φ

ϕ′ϕ′dx and S3 = 12

μ

H

∫ l+φ

φ

(ϕ′)2 dx,

H =∫ l+φ

φ

1/(1 + χA sin x)dx.

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

(3.15)

As in §3a, the requirement for the modulation of the beam stiffness to be small, χI � 1, isimposed for truncation of the solution series (3.1) to be adequate. Substituting the solution



10


...................................................

form (3.6) into terms multiplied by S3 in (3.14), one obtains expressions of the form H(bc) exp(iκx),which describe a wave with the same wavenumber κ and frequency ω as the primary one,but propagating in the opposite direction. Thus, the requirement for this additional wave to benegligibly weak and not affect the primary wave should be imposed for the analysis of the beamdispersion relation to be valid, leading to the condition

S3 � 1, (3.16)

which implies that the terms multiplied by S3 in (3.14) are much smaller than the leading terms,so that the solution form (3.6) can be employed.

Considering S1 and S2, it is found that for a relatively long beam l � 1, and with the solutionform (3.6) and the real-valued constants

�

bj and θj introduced according to (3.10), these can beapproximated as

S1 = μη1 − χ2

A + χ4A/8

1 − χ2A/2

(3.17)

and

S2 = μ

∞∑j=−∞

(κ − j)2 �

b2j

1 − χ2A + χ4

A/8

1 − χ2A/2

. (3.18)

The resulting algebraic equations for�

bj are

�

bj

((κ − j)4 − ω2

)+ (−1)j ω

2χA

2(

�

bj−1 − �

bj+1) + (−1)j (κ − j)2χI

2

(�

bj+1(κ − j − 1)2 − �

bj−1(κ − j + 1)2)

= −(κ − j)2 �

bj(S1 + S2), (3.19)

and θj satisfy relations (3.11). As can be seen (3.19) do not involve the length of the beam l andphase φ so that, as predicted (see §2b), the dispersion relation of the considered nonlinear beamdoes not depend on these parameters. The effect of the nonlinearity on this relation is governedby the term S2, present in (3.19). Comparing expressions (3.17), (3.18) for S1 and S2, it appearsthey differ only at the position of the initial pre-stretching coefficient η; hence, the nonlinearity isequivalent to an additional stretching of the beam

ηn =∞∑

j=−∞(κ − j)2 �

b2j , (3.20)

which depends on the magnitude of the beam transverse deflections w; for propagating waves(real values of κ) ηn > 0.

Thus, the approximate solution of the initial dimensionless equation (2.28) is obtained in theform (3.7), describing a propagating (or attenuating) wave with dimensionless frequency ω andwavenumber κ . As for case A, the amplitude B is introduced by (3.13) to define the magnitudesof

�

bj. Equations (3.19) are then solved numerically for�

bj and κ as functions of the amplitude Band parameters ω, χI, χA, μ and η.

4. Dispersion relations and frequency band-gaps

(a) Effects of nonlinear (true) curvature and nonlinear materialIn [21], it was shown that the linear dispersion relation of the considered periodic beam issymmetric about axis ω and periodic with respect to the wavenumber κ , which agrees wellwith the results obtained in the classical works [1,12]. For pure modulation χA of the beam massper unit length, the linear dispersion relation features two distinct band-gaps in the consideredfrequency range (at ω ≈ 0.25 and ω ≈ 1), whereas pure modulation χI of the beam stiffness givesone band-gap (at ω ≈ 0.25), and modulations with equal amplitudes χA = χI also one band-gap(at ω ≈ 1).



11


...................................................

1.5

1.0

0.5

k

w

k k0 0.5 1.0 0 0.5 1.0 0 0.5 1.0

(b)(a) (c)

Figure 2. Dispersion relations ω(κ ) for (a) linear material, βn = 0, and nonlinear curvature B= 0.45, χA = 0, χI = 0.5,(b) only the effect of nonlinear material taken into account,βn = 0.25 and B= 0.45,χA = 0.5,χI = 0, and (c) both sourcesof nonlinearity taken into account,βn = 0.15,B= 0.4,χA = χI = 0.5. Solid lines, nonlinear beam; dotted lines, linear beam.(Online version in colour.)

To simplify the analysis of the effects of nonlinearities on the dispersion relation, we firstconsider each source of nonlinearity separately: in this section, nonlinear (true) curvature andnonlinear material. The effects of nonlinear inertia are not taken into account, but will be studiedin §4c.

As follows from the obtained solution (3.7), the structure of the beam dispersion relationdoes not change owing to the nonlinearities, i.e. the relation remains symmetric and periodicwith respect to κ . The pure effect of nonlinear (true) curvature lies in shifting the dispersionrelation to higher frequencies, as is illustrated in figure 2a for B = 0.45, χA = 0, χI = 0.5 andβn = 0, with the linear dispersion relation shown for comparison (dotted line). The pure effectof nonlinear material is opposite to the effect of nonlinear curvature, i.e. the nonlinear dispersionrelation is shifted to lower frequencies (figure 2b). Consequently, these sources of nonlinearity cancompensate for the effect of each other, as is illustrated in figure 2c, where solid and dashed linesalmost coincide. The effects of nonlinearities appear to be more pronounced for higher frequenciesand the corresponding band-gaps.

According to the phase closure principle [30], the frequencies corresponding to the boundariesof band-gap regions for linear periodic structures are those where an integer number n ofcompound half-waves fits exactly into a unit cell of the structure, i.e. they correspond to thewavenumbers

κ = n2

, n = ±1, ±2, ±3, . . . . (4.1)

As follows from the obtained solution (3.7), which is of the same form as in the linear case andobeys Floquet theory, this holds also for the considered nonlinear periodic beam. So the criticalfrequencies ωc, determining boundaries of the band-gaps, can be obtained by letting κ = n/2,n = ±1, ±2, ±3, . . ., in (3.12). The effect of nonlinearities appears to be the same for differentvalues of the modulation amplitudes χA and χI, and is more pronounced for higher band-gaps. The widths of the band-gaps appear to be relatively insensitive to (weak) nonlinearities.As an illustration, figure 3a,b shows the dependencies of ωc corresponding to the first (n = 1)and the second (n = 2) band-gap on the amplitude B for βn = 0 and χA = 0.5, χI = 0. Figure 3c,drepresents these dependencies for the second band-gap with only the effect of nonlinear materialtaken into account for different modulation amplitudes χA and χI. Figure 3e corresponds to thecase of combined nonlinearities; as is seen the nonlinearities compensate for the effect of eachother. It is interesting to determine the critical value of the parameter βn at which there willbe complete compensation of the nonlinearities, i.e. the band-gap range will not change with



12


...................................................

0.301.10

1.05

1.05

1.00

0.95

1.051.30

1.15

1.00

1.00

0.95

1.00

0.26

0.22

0 0.2 0.4 0.6

0 0.2 0.4 0.6 0 0.2 0.4 0.6 0 0.2 0.4 0.6

0 0.2 0.4 0.6

B

B B

B B

wc

wc

(b)(a) (c)

(d ) (e)

Figure 3. Dependencies of the critical frequenciesωc , determining band-gap boundaries, on the amplitudeB for (a,b)βn = 0,χA = 0.5,χI = 0, (a) first band-gap, n= 1, (b) second band-gap, n= 2; (c,d) only the effect of nonlinearmaterial taken intoaccount,βn = 0.25, n= 2, (c)χA = 0.5,χI = 0, (d)χA = χI = 0.5; and (e) both sources of nonlinearity taken into account,n= 2,βn = 0.15,χA = 0.9,χI = 0. (Online version in colour.)

increasing amplitude B. Taking into account that the effect of nonlinearities depends weakly onthe modulation amplitudes χA and χI, we get

βncr ≈ 23n2 , (4.2)

where n is the number of the band-gap. As can be seen, it is possible to achieve completecompensation for the nonlinearities only for one of the frequency band-gaps, e.g. the second one,n = 2, in figure 3e for βncr ≈ 1

6 .

(b) Effects of initial pre-stretching and mid-plane stretching nonlinearityNext, consider the case when both ends of the beam are restricted from moving longitudinally(case B). First, we analyse the linear dispersion relation and the effects of initial pre-stretching η

of the beam. It appears that positive pre-stretching shifts the band-gaps to higher frequencies,and negative pre-stretching to lower frequencies, the effect being most pronounced for lowfrequencies. This is illustrated in figure 4a,b, with the dispersion relation without pre-stretchingbeing shown for comparison by dotted lines. In particular, it is possible to shift one of theboundaries of the lowest band-gap to zero frequency, with the width of the band-gap beingconsiderably increased (figure 4c). In this case, the beam serves as a high-pass filter, where allwaves with frequencies lower than a certain critical one are attenuated. Also, for χA = χA, whenthe unstretched beam does not feature a frequency band-gap at ω ≈ 0.25, the pre-stretched beamdoes (figure 4d).

The width of the second band-gap is relatively weakly affected by pre-stretching, though it canbe effectively shifted to a higher- or lower-frequency range. The width of the first band-gap, bycontrast, is strongly affected by pre-stretching, as is illustrated in figure 5, where the dependencies



13


...................................................

0 0.5 1.0k

0 0.5 1.0k

0 0.5 1.0k

0 0.5 1.0k

(b) (c) (d )1.5

1.0

0.5

w

(a)

Figure 4. Dispersion relationsω(κ ) for the linear pre-stretched (solid lines), and unstretched (dotted lines) beam forμ = 100and (a) η = 0.001, χA = 0.5, χI = 0; (b) η = −0.001, χA = 0, χI = 0.5; (c) η = −0.002, χA = 0, χI = 0.5; and (d)η = 0.002,χA = χI = 0.5. (Online version in colour.)

0.4

0.3

0.2

0.1

h–0.002 0.002 0.0050

h–0.002 0.002 0.0050

h–0.002 0.002 0.0050

wc

(b)(a) (c)

Figure 5. Dependency of the critical frequencies ωc , determining the boundaries of the first band-gap, on the initial pre-stretching η for μ = 100 and (a) χA = 0.5, χI = 0; (b) χA = 0, χI = 0.5; and (c) χA = χI = 0.5. (Online version incolour.)

of the critical frequencies ωc, determining the boundaries of the first band-gap, on the value of theinitial pre-stretching η are shown. For pure modulation of the beam mass per unit length, positivepre-stretching increases the band-gap, whereas negative pre-stretching decreases it, as seen infigure 5a, and at a certain value of η the width of the band-gap essentially vanishes. In the caseof pure modulation of the beam stiffness (figure 5b), the effect of pre-stretching is the opposite,and it is even possible to obtain a large band-gap with zero frequency as the lower boundary.If modulations with equal amplitudes are imposed (figure 5c), then negative as well as positivepre-stretching increases the band-gap.

Considering the isolated effect of mid-plane stretching nonlinearity, it is found that it is similarto that of nonlinear curvature: the band-gaps are shifted to higher frequencies, whereas the widthof the band-gaps is changed only slightly, as illustrated by figure 6. However, this source ofnonlinearity is much stronger than the nonlinear curvature, being pronounced already at verysmall values, B ∼ 10−2, of transverse beam deflections.

(c) Effects of nonlinear inertiaNow consider the isolated effect of nonlinear inertia, governed by the term (Nw′)′ in (2.14), onthe dispersion relation. This nonlinearity is involved in (2.14) along with nonlinear curvatureand nonlinear material; however, we discuss it separately, because the effects it causes differconsiderably from those already described in §4a.



14


...................................................

1.5

1.0

0.5

0 0.5 1.0 0 0.5 1.0 0 0.5 1.0k k k

w

(b)(a) (c)

Figure 6. Dispersion relationsω(κ ) for the nonlinear (solid lines) and linear (dotted lines) beam forμ = 100, B= 0.06 and(a) η = 0, χA = 0.5, χI = 0; (b) η = −0.001, χA = 0, χI = 0.5; and (c) η = 0.001, χA = χI = 0.5. (Online version incolour.)

Substituting the obtained solution wl for the linear beam problem into (Nw′)′, one finds thatfor integer values of 2κ and χA �= 0 or χI �= 0 the term tends to infinity for arbitrarily smallvalues of the amplitude B. Thus, the dispersion relation for the non-uniform beam should changeconsiderably for wavenumbers close to κ = 0, ± 1

2 , ±1, ± 32 , . . . owing to nonlinear inertia; it is

exactly at these wavenumbers that the frequency band-gaps arise in the linear case.Dispersion relations, relating a certain frequency to a certain wavenumber, as well as frequency

band-gaps are relevant for linear or weakly nonlinear wave motion only. For strongly nonlinearwaves, comprising many components with different frequencies, these notions are of little use [1,12]. For example, it is impossible or rather cumbersome to achieve attenuation of all componentsof such a wave by periodicity effects. In addition, when solving the initial equations by the MVA,the involved nonlinearities were assumed to be weak, so that only weakly nonlinear waves arecaptured. A 1 : 3 ratio between the maximum absolute values of the nonlinear term (Nw′)′ in (2.14)and the linear term [(1 + χI sin x)w′′]′′ has been chosen as a threshold to separate weakly nonlinearfrom strongly nonlinear wave motion.

The dispersion relation of the beam does not feature frequency band-gaps owing to thenonlinear inertia. Instead of the band-gaps, relatively narrow frequency ranges arise in whichthe wave motion is strongly nonlinear. As an illustration, figure 7 shows the dispersion relationof the considered beam with the isolated effect of nonlinear inertia taken into account, with thelinear dispersion relation shown for comparison by the dotted lines. Regions in which the wavemotion is strongly nonlinear are bounded by dashed lines; in these regions, nonlinear dispersionrelations are not shown, because the method employed, the MVA, captures only weakly nonlinearwaves. Figure 7a corresponds to the case of pure modulation of the beam mass per unit length,figure 7b to pure modulation of the beam stiffness, and figure 7c to modulations with equal non-zero amplitudes. As can be seen, frequency ranges with strongly nonlinear wave motion ariseinstead of the band-gaps and correspond to wavenumbers slightly shifted from κ = 0, ± 1

2 , ±1,± 3

2 , · · · . The size of the shift depends on the amplitude B: the larger the amplitude, the largerthe shift; compare, for example, figure 7e for B = 0.4 with figure 7f for B = 0.1. As appears fromfigure 7g,h, the effect of nonlinear inertia is strong even for very small beam deflections, B ∼ 10−2,so that the dispersion relation does not feature frequency band-gaps.

A frequency range implying a strongly nonlinear wave motion arises near κ = ±1/2 alsoin the case of modulations with equal amplitudes, χA = χI, when there is no band-gap in thelinear dispersion relation at κ = ±1/2 (figure 7h). The effects described above are present for thenon-uniform beam only; for the uniform beam, the influence of nonlinear inertia on the dispersion



15


...................................................

0.35

0.25

0.15

1.20.26

0.24

0.22

1.1

1.0

0.9

1.5

1.0

0.5

0

0.40

0.90 0.95 1.00 1.05 1.10

0.45 0.50 0.55 0.60 0.40

0.475 0.500 0.525

0.45 0.50 0.55 0.60

0.5 1.0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0

w

w

w

k k

(b)(a)

(g) (h)

(c) (d)

(e) ( f )

Figure 7. Dispersion relations ω(κ ) with only the effect of nonlinear inertia taken into account for (a–d) B= 0.4 and (a)χA = 0.5,χI = 0, (b)χA = 0,χI = 0.5, (c)χA = χI = 0.5, (d)χA = χI = 0.5; (e,f )χA = 0.5,χI = 0, and (e) B= 0.4,(f ) B= 0.1; (g,h) B= 0.02, χA = χI = 0.5. Solid lines, nonlinear beam; dotted lines, linear beam; dashed lines, regions inwhich the wave motion is strongly nonlinear. (Online version in colour.)

relation is much weaker (figure 7d). For example, for the wave motion to be strongly nonlinear,the beam deflections should be much larger than those considered previously, e.g. B ∼ 1.

The obtained results clearly indicate that nonlinear inertia has a substantial impact on the non-uniform beam dispersion relation. It appears to remove all the band-gaps, with the frequencyranges implying a strongly nonlinear wave motion arising instead. The effects of nonlinear inertiaseem to prevail over other nonlinearities considered in §4a, as illustrated by figure 8, which showsthe dispersion relation with all three sources of nonlinearity taken into account. The effects ofnonlinear inertia can be pronounced also in case B when both ends of the beam are restrictedfrom moving in the longitudinal direction, because these effects are strong even for very smallbeam deflections resulting in vanishing of all the band-gaps.

From the results obtained, it follows that real periodic beam structures with continuousmodulations of parameters performing bending oscillations should not feature frequencyband-gaps. In the case of piecewise constant modulations, however, the effects of nonlinearinertia can be much weaker, as is suggested by the results obtained for the uniform beam.So such beams can feature frequency band-gaps, which has also been shown by laboratoryexperiments [12].



16


...................................................

1.5

1.0

0.5

0 0.5 1.0k

0 0.5 1.0k

0 0.5 1.0k

w

(b)(a) (c)

Figure 8. Dispersion relations ω(κ ) with the effects of nonlinear inertia, nonlinear curvature and nonlinear material takeninto account for B= 0.4 and (a) βn = 0.25, χA = 0.5, χI = 0; (b) βn = 0.2, χA = 0, χI = 0.5; (c) βn = 0.3, χA = 0.5,χI = 0.5. Solid lines, nonlinear beam; dotted lines, linear beam. (Online version in colour.)

5. Validation of the results

(a) Numerical validationAs appears from §§2 and 3, the obtained analytical solution involves two approximations. Thefirst one is concerned with the truncation of the series (2.16), and the second is implied in the MVA.Both of them imply certain nonlinear high-order harmonic terms in the equations considered tobe discarded. This simplification is valid under the condition that the involved nonlinearities areweak, which is the key assumption of this analysis, and has been carefully checked for all resultspresented.

For the isolated effects of the nonlinear curvature and nonlinear material, this conditionappears to be satisfied if

B2 � 1, (5.1)

for κ = O(1) and βn = O(1). Relation (5.1) was considered to be fulfilled for B2 ≤ 1/3. Mid-planestretching nonlinearity appears to be weak if the following condition is satisfied:

12 μB2 � 1, (5.2)

which was considered to be fulfilled for 12 μB2 ≤ 1

3 .By contrast to the other sources of nonlinearity, nonlinear inertia implies strongly nonlinear

wave motion in certain, relatively narrow, frequency ranges for any, even very small, B ∼ 10−3,beam deflections (see §4c). A 1 : 3 ratio between the maximum absolute values of the nonlinearterm (Nw′)′ in (2.14) and the linear term

[(1 + χI sin x) w′′]′′ has been employed to define the

boundaries of these frequency ranges, i.e. it was chosen as a threshold to separate weaklynonlinear from strongly nonlinear wave motion. In figure 7, the regions with strongly nonlinearwave motion are bounded by dashed lines.

The MVA implies also that certain linear terms are discarded; as is shown, this is valid undercondition (3.5), which leads to the requirement for modulations of the beam stiffness to be small,χI � 1. To further validate the results, a series of numerical experiments were conducted. Theinitial non-dimensional governing equation (2.14) (or (2.25)) was numerically integrated directly



17


...................................................

using Wolfram MATHEMATICA v. 7.0 (NDSolve), with periodic boundary conditions and thefollowing initial conditions imposed:

w(x, 0) = 2 [(�

b0 + (�

b11 − �

b12) sin x + (�

b22 + �

b21) cos 2x) cos κx

− ((�

b11 + �

b12) cos x + (�

b22 − �

b21) sin 2x) sin κx]

and∂w∂t

(x, 0) = 2ω [(�

b0 + (�

b11 − �

b12) sin x + (�

b22 + �

b21) cos 2x) sin κx

+ ((�

b11 + �

b12) cos x + (�

b22 − �

b21) sin 2x) cos κx] .

∣∣∣∣∣∣∣∣∣∣∣∣∣(5.3)

These initial conditions correspond to the obtained analytical solution (3.7) for θ = 0 and the firstfive terms taken into account in series (3.1). Consequently, in accordance with the theoreticalpredictions, at such initial conditions, the beam should oscillate with frequency ω. This allowsthe obtained dispersion relations between frequency ω and wavenumber κ , as well as thesolution (3.7) itself, to be validated.

Typical results of the numerical experiments are shown in figure 9, where solid lines are valuesof the frequency ω obtained analytically (see §4), and filled circles represent numerical data forvarious values of the modulation amplitudes χI, χA and other parameters.

Figure 9a illustrates the pure effect of nonlinear curvature for B = 0.5, χA = 0.5, χI = 0, andfigure 9b the pure effect of nonlinear material for βn = 0.25, B = 0.5, χA = χI = 0.5. The discrepancybetween the numerical and analytical values of the frequency ω is less than 0.3% for all values ofκ , although in case (b) for κ = 1 it rises to 1%. Additional (high-)frequency components are presentin the beam response; these are due to the nonlinearity, and at the parameter values considered donot exceed 3% of the total response amplitude; the larger the frequency ω, the more pronouncedthese components become.

Figure 9c illustrates the effect of initial pre-stretching of the linear beam for μ = 100, η = −0.002,χA = 0, χI = 0.5. Here, the discrepancy between the numerical and analytical values of thefrequency ω is even smaller, around 0.2%, and no additional frequency components are present inthe beam response. Figure 9d represents mid-plane stretching nonlinearity for μ = 100, B = 0.06,η = 0.001, χA = χI = 0.5. It should be noted that Wolfram MATHEMATICA, as well as other similarsoftware packages, is not able to handle nonlinear partial integrodifferential equations. Thus,when solving numerically the considered equation, the integral term was calculated using theobtained analytical solution. The resulting discrepancy between the numerical and analyticalvalues of the frequency ω is less than 0.5%, and the additional frequency components in the beamresponse do not exceed 5% of the total response amplitude. To validate the simplification that weemployed, the integral term was calculated using the obtained numerical solution and comparedwith the analytical one; the resulting discrepancy between them was less than 0.6%.

Figure 9e,g illustrates the isolated effect of nonlinear inertia for B = 0.4, χA = 0.5, χI = 0. Herethe discrepancy between numerical and analytical values of the frequency ω is again very small,around 0.2%. However, the additional (high-)frequency components in the beam response areabout 10% for κ near 0.5 and 1, and the closer the wavenumber κ to the regions in which the wavemotion is strongly nonlinear, the more pronounced these components become. Similar resultswere obtained for the case when all three sources of the nonlinearity were taken into account:figure 9f ,h for B = 0.4 and (f ) βn = 0.2, χA = 0, χI = 0.5; (h) βn = 0.3, χA = χI = 0.5.

In summary, good agreement between the numerical and analytical results for all theconsidered cases can be noted, and, thus, the effects revealed in §4 are validated numerically.

(b) Experimental validationTo further validate the results obtained, laboratory experiments [34] were conducted. Two 1 mlong steel beams with rectangular cross section of a constant height h and varying width b wereimpacted by an instrumented impact hammer, and their frequency response functions (FRFs)thus determined and band-gaps identified. The height of the first beam was h = 5 mm and thatof the second beam was h = 15 mm. The width of both beams varied piecewise linearly (so



18


...................................................

1.5

1.0

0.5

1.5

1.0

0.5

0.351.50

1.25

1.00

0.75

0.25

0.15

0 0.5

0.40 0.45 0.50 0.55 0.60 0.8 0.9 1.0 1.1 1.2

1.0 0 0.5 1.0 0 0.5 1.0

k k

w

w

w

(b)(a) (c)

(d )

(g) (h)

(e) ( f )

Figure 9. Dispersion relations ω(κ ) illustrate (a) the isolated effect of nonlinear curvature B= 0.5, χA = 0.5, χI = 0; (b)the isolated effect of nonlinear material βn = 0.25, B= 0.5, χA = χI = 0.5; (c) the isolated effect of initial pre-stretchingof the linear beam μ = 100, η = −0.002, χA = 0, χI = 0.5; (d) the isolated effect of mid-plane stretching nonlinearityμ = 100, B= 0.06, η = 0.001, χA = χI = 0.5; (e,g) the isolated effect of nonlinear inertia B= 0.4, χA = 0.5, χI = 0;(f ,h) the combined effect of nonlinear inertia, nonlinear curvature and nonlinear material B= 0.4 and (g)βn = 0.2,χA = 0,χI = 0.5; (h)βn = 0.3,χA = χI = 0.5. Solid lines, analytical results; filled circles, numerical data. (Online version in colour.)

that the considered modulation was continuous), as is shown in figure 10a, with bmin = 25 andbmax = 75 mm.

A sketch of the experimental set-up is shown in figure 10b. The beam hangs in soft rubberbands which simulate free–free boundary conditions. An accelerometer at beam point 6 monitorsacceleration in the y-direction (figure 10c). FRFs were obtained by exciting the structure at points1–5 by the impact hammer and measuring the output at point 6. Each of the measurementswas repeated three times, checking for acceptable measures of coherence (close to unity in thefrequency range of relevance except at (anti-)resonance).

The frequency spectrum of the measured force at perfect impact is constant, providingexcitation energy at all frequencies. A real hammer hit, however, features a finite frequency rangeof excitation, which depends on the stiffness of the tip of the hammer and the excited structure;in the experiments, the excited frequency range was tailored to be below about 5 kHz.



19


...................................................

forcetransducerB&K 8200

impacthammer

accelerometerB&K 4397

B&K PULSEanalyser software+modal testconsultant

ME' scopemodal analysissoftware

LANswitch

LAN

PC

input/outputfront-endB&K 3160-A-4/2

1 2

chargeconverterB&K 2646

y

z

y

z

1 2 3 4 5 6

x~

x~

(b)

(a)

(c)

Figure 10. (a) Experimental beam with the height h= 15 mm. Impulse from the impact hammer is applied at points 1–5;(b) experimental set-up; and (c) beam support and accelerometer mounted at point 6. (Adapted from [34].) (Online version incolour.)

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Figure 11. Experimental FRFs of (a) the first beam (h= 5 mm) and (b) the second beam (h= 15 mm). (Adapted from [34].)

The results of the experiments are shown in figure 11; (a) corresponds to the beam with h =5 mm and (b) to the beam with h = 15 mm. No frequency band-gaps can be detected from theFRF diagrams; this supports the conclusions of §4, where it was shown that real periodic beamstructures with continuous modulations of parameters performing bending oscillations should notfeature frequency band-gaps owing to nonlinear inertia. The linear theory predicts band-gaps in



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the considered frequency range, e.g. for the first beam the first band-gap should be near 1.1 kHz,and for the second beam this band-gap is from 3.3 to 3.4 kHz, indicating that this theory does notsuffice to correctly describe the beams’ response and nonlinearities should be accounted for, as isdone in the theoretical part of this paper.

The results presented in this section may be considered as a first step in the experimentalvalidation of the theoretical predictions of the paper. Further in-depth experimental testing isrelevant, e.g. considering periodic beams with wider frequency band-gaps predicted by thelinear theory.

6. ConclusionThe effects of weak nonlinearity on the dispersion relation and frequency band-gaps of a periodicBernoulli–Euler beam performing bending oscillations are analysed. Two cases are considered:(i) relatively large transverse deflections, where nonlinear (true) curvature, nonlinear materialand nonlinear inertia owing to longitudinal motions of the beam are taken into account, and (ii)mid-plane stretching nonlinearity. As a result, several notable effects are revealed by means ofthe method of varying amplitudes; in particular, a shift of the band-gaps to a higher frequencyowing to nonlinear curvature, whereas the effect of nonlinear material is the opposite. The widthof the band-gaps appears to be relatively insensitive to these nonlinearities. It is shown that initialpre-stretching of the beam considerably affects the dispersion relation: it is possible for new band-gaps to emerge and the band-gaps can be shifted to a higher or lower frequency, their widthbeing considerably changed. The isolated effects of mid-plane stretching nonlinearity are similarto those of nonlinear curvature, though mid-plane stretching nonlinearity is pronounced alreadyat much smaller beam deflections.

It has been shown that, of the four sources of nonlinearity considered, nonlinear inertia hasthe most substantial impact on the dispersion relation of a non-uniform beam with continuousmodulations of cross-section parameters. It appears to remove all the band-gaps, with thefrequency ranges implying a strongly nonlinear wave motion arising instead. The results obtainedare validated by experiments and numerical simulation, and explanations of the revealed effectsare suggested.

Data accessibility. This article contains no external data.Authors’ contributions. V.S.S. derived the governing equations and performed their solution by the MVA,found the nonlinear dispersion relations and conducted numerical validation of the obtained results. J.J.T.supervised the research and established the logical organization of the paper.Competing interests. We declare we have no competing interests.Funding. The work was carried out with financial support from the Danish Council for Independent Researchand FP7 Marie Curie Actions—COFUND: DFF—1337-00026.Acknowledgements. The authors are grateful to Prof. J. S. Jensen and Corresponding Member of the RussianAcademy of Science D. A. Indeitsev for valuable comments on the paper, and to Solveig Dadadottir for herefforts in conducting the presented laboratory experiments.

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